Estimation of convolution in the model with noise

Christophe Chesneau 1 Fabienne Comte 2 Gwennaelle Mabon 2 Fabien Navarro 3
1 LMNO, Université de Caen Basse-Normandie
LMNO - Laboratoire de Mathématiques Nicolas Oresme
3 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : We investigate the estimation of the $\ell$-fold convolution of the density of an unobserved variable $X$ from $n$ i.i.d. observations of the convolution model $Y=X+\varepsilon$. We first assume that the density of the noise $\varepsilon$ is known and define nonadaptive estimators, for which we provide bounds for the mean integrated squared error (MISE). In particular, under some smoothness assumptions on the densities of $X$ and $\varepsilon$, we prove that the parametric rate of convergence $1/n$ can be attained. Then we construct an adaptive estimator using a penalization approach having similar performances to the nonadaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.
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Article dans une revue
Journal of Nonparametric Statistics, American Statistical Association, 2015, 27 (3), pp.286-315
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Christophe Chesneau, Fabienne Comte, Gwennaelle Mabon, Fabien Navarro. Estimation of convolution in the model with noise. Journal of Nonparametric Statistics, American Statistical Association, 2015, 27 (3), pp.286-315. <hal-01003005>

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